Difference between revisions of "2005 Alabama ARML TST Problems/Problem 10"
(New page: ==Problem== When Jon Stewart walks up stairs he takes one or two steps at a time. His stepping sequence is not necessarily regular. He might step up one step, then two, then two again, the...) |
(→Solution) |
||
| Line 5: | Line 5: | ||
He can either take 14 one-steps, or 12 one-steps and 1 two-step, etc., so we have | He can either take 14 one-steps, or 12 one-steps and 1 two-step, etc., so we have | ||
| − | <math>\frac{14!}{14!}+\frac{13!}{12!\cdot 1!}+\frac{12!}{10!\cdot 2!}+\frac{11!}{8!\cdot 3!}+\frac{10!}{6!\cdot 4!}+\frac{9!}{4!\cdot 5!}+\frac{8!}{2!\cdot 6!}+\frac{7!}{7!}=1+13+66+165+210+126+28+1=14+231+336+29=245+365=\boxed{610}</math> | + | <math>\begin{eqnarray} |
| + | \frac{14!}{14!}+\frac{13!}{12!\cdot 1!}+\frac{12!}{10!\cdot 2!}+\frac{11!}{8!\cdot 3!}+\frac{10!}{6!\cdot 4!}+\frac{9!}{4!\cdot 5!}+\frac{8!}{2!\cdot 6!}+\frac{7!}{7!}\\ | ||
| + | =1+13+66+165+210+126+28+1=14+231+336+29=245+365=\boxed{610} | ||
| + | \end{eqnarray}</math> | ||
==See also== | ==See also== | ||
Revision as of 16:30, 1 March 2008
Problem
When Jon Stewart walks up stairs he takes one or two steps at a time. His stepping sequence is not necessarily regular. He might step up one step, then two, then two again, then one, then one, and then two in order to climb up a total of 9 steps. In how many ways can Jon walk up a 14 step stairwell?
Solution
He can either take 14 one-steps, or 12 one-steps and 1 two-step, etc., so we have
$\begin{eqnarray} \frac{14!}{14!}+\frac{13!}{12!\cdot 1!}+\frac{12!}{10!\cdot 2!}+\frac{11!}{8!\cdot 3!}+\frac{10!}{6!\cdot 4!}+\frac{9!}{4!\cdot 5!}+\frac{8!}{2!\cdot 6!}+\frac{7!}{7!}\\ =1+13+66+165+210+126+28+1=14+231+336+29=245+365=\boxed{610} \end{eqnarray}$ (Error compiling LaTeX. Unknown error_msg)
